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FAQ: What are the differences between one-tailed and two-tailed tests?

When you conduct a test of statistical significance, whether it is from a correlation, an ANOVA, a regression or some other kind of test, you are given a p-value somewhere in the output.  If your test statistic is symmetrically distributed, you can select one of three alternative hypotheses. Two of these correspond to one-tailed tests and one corresponds to a two-tailed test.  However, the p-value presented is (almost always) for a two-tailed test.  But how do you choose which test?  Is the p-value appropriate for your test? And, if it is not, how can you calculate the correct p-value for your test given the p-value in your output?  

What is a two-tailed test?

First let’s start with the meaning of a two-tailed test.  If you are using a significance level of 0.05, a two-tailed test allots half of your alpha to testing the statistical significance in one direction and half of your alpha to testing statistical significance in the other direction.  This means that .025 is in each tail of the distribution of your test statistic. When using a two-tailed test, regardless of the direction of the relationship you hypothesize, you are testing for the possibility of the relationship in both directions.  For example, we may wish to compare the mean of a sample to a given value x using a t-test.  Our null hypothesis is that the mean is equal to x . A two-tailed test will test both if the mean is significantly greater than x and if the mean significantly less than x . The mean is considered significantly different from x if the test statistic is in the top 2.5% or bottom 2.5% of its probability distribution, resulting in a p-value less than 0.05.     

What is a one-tailed test?

Next, let’s discuss the meaning of a one-tailed test.  If you are using a significance level of .05, a one-tailed test allots all of your alpha to testing the statistical significance in the one direction of interest.  This means that .05 is in one tail of the distribution of your test statistic. When using a one-tailed test, you are testing for the possibility of the relationship in one direction and completely disregarding the possibility of a relationship in the other direction.  Let’s return to our example comparing the mean of a sample to a given value x using a t-test.  Our null hypothesis is that the mean is equal to x . A one-tailed test will test either if the mean is significantly greater than x or if the mean is significantly less than x , but not both. Then, depending on the chosen tail, the mean is significantly greater than or less than x if the test statistic is in the top 5% of its probability distribution or bottom 5% of its probability distribution, resulting in a p-value less than 0.05.  The one-tailed test provides more power to detect an effect in one direction by not testing the effect in the other direction. A discussion of when this is an appropriate option follows.   

When is a one-tailed test appropriate?

Because the one-tailed test provides more power to detect an effect, you may be tempted to use a one-tailed test whenever you have a hypothesis about the direction of an effect. Before doing so, consider the consequences of missing an effect in the other direction.  Imagine you have developed a new drug that you believe is an improvement over an existing drug.  You wish to maximize your ability to detect the improvement, so you opt for a one-tailed test. In doing so, you fail to test for the possibility that the new drug is less effective than the existing drug.  The consequences in this example are extreme, but they illustrate a danger of inappropriate use of a one-tailed test.

So when is a one-tailed test appropriate? If you consider the consequences of missing an effect in the untested direction and conclude that they are negligible and in no way irresponsible or unethical, then you can proceed with a one-tailed test. For example, imagine again that you have developed a new drug. It is cheaper than the existing drug and, you believe, no less effective.  In testing this drug, you are only interested in testing if it less effective than the existing drug.  You do not care if it is significantly more effective.  You only wish to show that it is not less effective. In this scenario, a one-tailed test would be appropriate. 

When is a one-tailed test NOT appropriate?

Choosing a one-tailed test for the sole purpose of attaining significance is not appropriate.  Choosing a one-tailed test after running a two-tailed test that failed to reject the null hypothesis is not appropriate, no matter how "close" to significant the two-tailed test was.  Using statistical tests inappropriately can lead to invalid results that are not replicable and highly questionable–a steep price to pay for a significance star in your results table!   

Deriving a one-tailed test from two-tailed output

The default among statistical packages performing tests is to report two-tailed p-values.  Because the most commonly used test statistic distributions (standard normal, Student’s t) are symmetric about zero, most one-tailed p-values can be derived from the two-tailed p-values.   

Below, we have the output from a two-sample t-test in Stata.  The test is comparing the mean male score to the mean female score.  The null hypothesis is that the difference in means is zero.  The two-sided alternative is that the difference in means is not zero.  There are two one-sided alternatives that one could opt to test instead: that the male score is higher than the female score (diff  > 0) or that the female score is higher than the male score (diff < 0).  In this instance, Stata presents results for all three alternatives.  Under the headings Ha: diff < 0 and Ha: diff > 0 are the results for the one-tailed tests. In the middle, under the heading Ha: diff != 0 (which means that the difference is not equal to 0), are the results for the two-tailed test. 

Two-sample t test with equal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- male | 91 50.12088 1.080274 10.30516 47.97473 52.26703 female | 109 54.99083 .7790686 8.133715 53.44658 56.53507 ---------+-------------------------------------------------------------------- combined | 200 52.775 .6702372 9.478586 51.45332 54.09668 ---------+-------------------------------------------------------------------- diff | -4.869947 1.304191 -7.441835 -2.298059 ------------------------------------------------------------------------------ Degrees of freedom: 198 Ho: mean(male) - mean(female) = diff = 0 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 t = -3.7341 t = -3.7341 t = -3.7341 P < t = 0.0001 P > |t| = 0.0002 P > t = 0.9999

Note that the test statistic, -3.7341, is the same for all of these tests.  The two-tailed p-value is P > |t|. This can be rewritten as P(>3.7341) + P(< -3.7341).  Because the t-distribution is symmetric about zero, these two probabilities are equal: P > |t| = 2 *  P(< -3.7341).  Thus, we can see that the two-tailed p-value is twice the one-tailed p-value for the alternative hypothesis that (diff < 0).  The other one-tailed alternative hypothesis has a p-value of P(>-3.7341) = 1-(P<-3.7341) = 1-0.0001 = 0.9999.   So, depending on the direction of the one-tailed hypothesis, its p-value is either 0.5*(two-tailed p-value) or 1-0.5*(two-tailed p-value) if the test statistic symmetrically distributed about zero. 

In this example, the two-tailed p-value suggests rejecting the null hypothesis of no difference. Had we opted for the one-tailed test of (diff > 0), we would fail to reject the null because of our choice of tails. 

The output below is from a regression analysis in Stata.  Unlike the example above, only the two-sided p-values are presented in this output.

Source | SS df MS Number of obs = 200 -------------+------------------------------ F( 2, 197) = 46.58 Model | 7363.62077 2 3681.81039 Prob > F = 0.0000 Residual | 15572.5742 197 79.0486001 R-squared = 0.3210 -------------+------------------------------ Adj R-squared = 0.3142 Total | 22936.195 199 115.257261 Root MSE = 8.8909 ------------------------------------------------------------------------------ socst | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- science | .2191144 .0820323 2.67 0.008 .0573403 .3808885 math | .4778911 .0866945 5.51 0.000 .3069228 .6488594 _cons | 15.88534 3.850786 4.13 0.000 8.291287 23.47939 ------------------------------------------------------------------------------

For each regression coefficient, the tested null hypothesis is that the coefficient is equal to zero.  Thus, the one-tailed alternatives are that the coefficient is greater than zero and that the coefficient is less than zero. To get the p-value for the one-tailed test of the variable science having a coefficient greater than zero, you would divide the .008 by 2, yielding .004 because the effect is going in the predicted direction. This is P(>2.67). If you had made your prediction in the other direction (the opposite direction of the model effect), the p-value would have been 1 – .004 = .996.  This is P(<2.67). For all three p-values, the test statistic is 2.67. 

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One-Tail vs Two-Tail Tests

Two-tailed Test

When testing a hypothesis, you must determine if it is a one-tailed or a two-tailed test. The most common format is a two-tailed test, meaning the critical region is located in both tails of the distribution. This is also referred to as a non-directional hypothesis.

This type of test is associated with a "neutral" alternative hypothesis. Here are some examples:

• There is a difference between the scores.
• The groups are not equal .
• There is a relationship between the variables.

One-tailed Test

The alternative option is a one-tailed test. As the name implies, the critical region lies in only one tail of the distribution. This is also called a directional  hypothesis. The image below shows a right-tailed test. A left-tailed test would be another type of one-tailed test.

This type of test is associated with a more specific alternative claim. Here are some examples:

• One group is higher than the other.
• There is a decrease in performance.
• Group A performs worse than Group B.
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4 One-tailed vs two-tailed test

To gain a deeper understanding of how to conduct a hypothesis test, this section will delve into the concepts of one-tailed and two-tailed tests. These tests are vital tools in statistical hypothesis testing, and the decision of which test to employ depends on the research question and hypothesis under examination. It is crucial to give careful thought to the suitable type of test to ensure that the hypothesis is thoroughly tested and precise conclusions are derived from the data. This section will elaborate on this topic in greater detail.

To commence, complete the following activity pertaining to the formulation of null and alternative hypotheses. This exercise may be somewhat challenging, but it serves as an excellent introduction to upcoming discussions – don’t be concerned if you find it difficult!

Activity 3 Hypotheses setting

Read the following statements and then develop a null hypothesis and an alternative hypothesis.

‘It is believed that OU students need to set aside no longer than, on average, 15 hours to study an entire session of an OU course. However, a researcher believes that OU students spend longer studying an entire session of an OU course.’

H 0 : OU students spend, on average, no more than 15 hours studying an entire session of OU course.

H a : OU students spend, on average, more than 15 hours studying an entire session of OU course.

They can also be written as:

H 0 : µ ≤ 15 hours studies

H a : µ > 15 hours studies

µ is a symbol for a population mean. Remember, H 0 and H a are always opposites.

Did you identify any differences between the hypotheses you developed in Activity 1 and Activity 3? The set of hypotheses in Activity 1 has an equal (=) or not equal (≠) supposition (sign) in the statement. However, in Activity 3, the set of hypotheses has less than or equal to (≤) and greater than (>) supposition (sign) in the statement. This creates different conditions that lead to acceptance or rejection of the null hypothesis.

One and Two Tailed Tests

Suppose we have a null hypothesis H 0 and an alternative hypothesis H 1 . We consider the distribution given by the null hypothesis and perform a test to determine whether or not the null hypothesis should be rejected in favour of the alternative hypothesis.

There are two different types of tests that can be performed. A one-tailed test looks for an increase or decrease in the parameter whereas a two-tailed test looks for any change in the parameter (which can be any change- increase or decrease).

We can perform the test at any level (usually 1%, 5% or 10%). For example, performing the test at a 5% level means that there is a 5% chance of wrongly rejecting H 0 .

If we perform the test at the 5% level and decide to reject the null hypothesis, we say "there is significant evidence at the 5% level to suggest the hypothesis is false".

One-Tailed Test

We choose a critical region. In a one-tailed test, the critical region will have just one part (the red area below). If our sample value lies in this region, we reject the null hypothesis in favour of the alternative.

Suppose we are looking for a definite decrease. Then the critical region will be to the left. Note, however, that in the one-tailed test the value of the parameter can be as high as you like.

Suppose we are given that X has a Poisson distribution and we want to carry out a hypothesis test on the mean, l, based upon a sample observation of 3.

Suppose the hypotheses are: H 0 : l = 9 H 1 : l < 9

We want to test if it is "reasonable" for the observed value of 3 to have come from a Poisson distribution with parameter 9. So what is the probability that a value as low as 3 has come from a Po(9)?

P(X < 3) = 0.0212 (this has come from a Poisson table)

The probability is less than 0.05, so there is less than a 5% chance that the value has come from a Poisson(3) distribution. We therefore reject the null hypothesis in favour of the alternative at the 5% level.

However, the probability is greater than 0.01, so we would not reject the null hypothesis in favour of the alternative at the 1% level.

Two-Tailed Test

In a two-tailed test, we are looking for either an increase or a decrease. So, for example, H 0 might be that the mean is equal to 9 (as before). This time, however, H 1 would be that the mean is not equal to 9. In this case, therefore, the critical region has two parts:

Lets test the parameter p of a Binomial distribution at the 10% level.

Suppose a coin is tossed 10 times and we get 7 heads. We want to test whether or not the coin is fair. If the coin is fair, p = 0.5 . Put this as the null hypothesis:

H 0 : p = 0.5 H 1 : p =(doesn' equal) 0.5

Now, because the test is 2-tailed, the critical region has two parts. Half of the critical region is to the right and half is to the left. So the critical region contains both the top 5% of the distribution and the bottom 5% of the distribution (since we are testing at the 10% level).

If H 0 is true, X ~ Bin(10, 0.5).

If the null hypothesis is true, what is the probability that X is 7 or above? P(X > 7) = 1 - P(X < 7) = 1 - P(X < 6) = 1 - 0.8281 = 0.1719

Is this in the critical region? No- because the probability that X is at least 7 is not less than 0.05 (5%), which is what we need it to be.

So there is not significant evidence at the 10% level to reject the null hypothesis.

One-tailed and Two-tailed Tests

by Karen Grace-Martin   30 Comments

I was recently asked about when to use one and two tailed tests.

The long answer is:  Use one tailed tests when you have a specific hypothesis about the direction of your relationship.  Some examples include you hypothesize that one group mean is larger than the other; you hypothesize that the correlation is positive; you hypothesize that the proportion is below .5.

The short answer is: Never use one tailed tests.

1. Only a few statistical tests even can have one tail: z tests and t tests.  So you’re severely limited.  F tests, Chi-square tests , etc. can’t accommodate one-tailed tests because their distributions are not symmetric.  Most statistical methods, such as regression and ANOVA , are based on these tests, so you will rarely have the chance to implement them.

2. Probably because they are rare, reviewers balk at one-tailed tests.  They tend to assume that you are trying to artificially boost the power of your test.  Theoretically, however, there is nothing wrong with them when the hypothesis and the statistical test are right for them.

Reader Interactions

June 10, 2022 at 5:48 am

Dear Karen,

I am running a regression analysis, I wonder if the significance in table output for ANOVA is inherently one-tailed or two-tailed? I have a directional hypothesis, so I am wondering if I still have to divide the significance value? Thank you in advance!

June 15, 2022 at 11:08 am

It’s two-tailed. And the F tests cannot be made directional.

October 9, 2019 at 7:46 pm

Say for example I am testing a hypothesis at .05 and it is a one tailed test. I expect the treatment to have a decrease and the cut off for significance is t = -1.67 so this is a left tail test. If t value after I run the test is +2.24 is this non significant since it is not in the hypothesized direction (negative)?

August 30, 2019 at 2:13 am

does a one-tailed test always require a one-sided (directional) hypothesis, and does a two-tailed test always require a two-sided (nondirectional) hypothesis?

Asked in the other direction: Does a one-sided (directional) hypothesis always require a one-sided test, and does a two-sided (nondirectional) hypothesis always require a two-sided test?

Thanks for clarifying Claudia

September 3, 2019 at 10:11 am

Hi Claudia,

Good question. So technically, when I am talking about one-tailed test, I do mean a test of a directional hypothesis. Likewise, I mean a non-directional hypothesis when I say two-tailed test.

In z-tests and t-tests, which are symmetric distributions, these terms are indeed interchangeable. But it definitely gets more complicated once you start talking about Chi-square tests and F-tests. Both of these have 0 at the left end and all high values of the statistic in one tail. So you can’t distinguish directions for these kinds of test statistics. So you can only test nondirectional hypotheses.

May 1, 2019 at 2:30 am

I have seen textbooks reporting confidence intervals of a standard deviation using low and high values of the chi-square statistic. Is it not based on a two-sided test?

August 28, 2018 at 7:14 am

There is no need to have a specific directional hypothesis (although you would usually have one), all that is needed to justify is to have a directional claim. Most claims in published scientific research and applied research are directional, since the moment you say the difference is positive or negative you have a directional claim. The only way to avoid it is to not mention the observed difference, or to state the difference as “plus or minus X” which would be ridiculous in most contexts.

Also, you should not mistake the tailed-ness of a statistical distribution with the tailed-ness of a hypothesis. While the Chi-Square or F-distribution might only have one tail, they can still be used for inference of one-sided and two-sided hypothesis alike. One can go as far back as Fisher and find examples of that (Statistical Methods for Research Workers).

If you are interested in more detailed arguments for the use of one-sided tests see the series of articles on The One-Sided Project website at https://www.onesided.org/ .

March 2, 2018 at 1:07 am

Hello, Good Morning to all, I am working on a project of Behavior Based Safety thesis work in which I have to assess data by SPSS software. Can any one could help me because i haven’t any idea about this software

May 17, 2018 at 10:06 am

Hi Muhammad, We have a number of free resources on SPSS, but if you’re in the middle of a project we have an on-demand tutorial that will get you not just started, but able to use it: http://theanalysisinstitute.com/introspss/

August 24, 2016 at 12:08 pm

Thank you for the discussions about the power for one-sided test. I agree that we should be careful when we decide to use a one-sided test. For Chi-square test when comparing two proportions, we can use two approaches: normal-theory method (the z-test) and contingency-table approach (the Chi-square test). For the normal-theory test, it requires a large sample size with n>5 or n*proportion >10. If your proposed study satisfied this requirement, we can use normal-theory method which is z-test to compute the power for the one-sided test. If you are interested in how to compute the one-sided test power, you can refer to the textbook by Marcello Pagano and Kimberlee Gauvreau’s 2nd edition titled “Principles of Biostatistics” section 14.5 on page 330 regarding sample size estimation for one-sided hypothesis test for proportions. Thanks.

September 1, 2015 at 2:30 am

Chi-squared tests are ALWAYS one-tailed… You only reject the null hypothesis when the test statistic falls into the right tail of the chi-square distribution.

What chi-squared tests are generally NOT is “directional”. They generally do not test whether the observed values are greater than or smaller than expected, only that they differ significantly.

January 26, 2016 at 10:57 pm

If “Chi-squared tests are ALWAYS one-tailed…”, why is Karen saying in her above statement that “F tests, Chi-square tests, etc. can’t accommodate one-tailed tests”?

Sorry, but I am bit confused. Can you help?

January 27, 2016 at 11:30 am

Hi Dippies,

Perhaps better wording is “F tests, Chi-square tests, etc. can’t accommodate directional tests.” Because there is only one tail for these distributions in which to find significance, it can’t distinguish between non-directional tests (eg, H1: mu1 – mu2 not equal to 0) and directional tests (eg, H1: mu1-mu2 greater than 0). In a t-test or z-test, we can either split alpha between two tails for a non-directional test or put alpha all into one tail for a directional test. We can then see whether we’re in the right tail based on the sign of the test statistic. F’s and Chi-sq values are squared. So they’re always positive. You get the same F value regardless of the direction of the means.

October 19, 2017 at 10:29 am

Thanks for the explanation… I am wondering: what happens if I have two methods, m1 and m2, and I want to show that m1 performs better (e.g. gives a higher value) than m2? Should I still use a two-tailed test? How can I show that m1 is, in fact, better (and not just different) than m2 (assuming the test proves significant)? Thanks for any comments.

July 22, 2015 at 6:14 pm

My hypothesis says there is a positive relation between religiosity and altruistic behavior. Would a two-tailed approach using Pearson Correlation do the trick?

Some very helpful information available here! Thank you.

July 1, 2015 at 2:50 am

Hi Karen, I run paired sample t-test and it just has P-two tailed. So how to convert P-two tailed to t critical one tailed. I have t critical two tailed and df already. Thanks so much

March 20, 2015 at 2:41 pm

I have used a one-tailed test but the effect went into the opposite direction. How do I have to calculate my p-Value now. Thanks for your reply, Mirco

February 3, 2015 at 10:24 pm

F-tests are almost always one-tailed. You would convert a two-tailed test’s p-value into a one-tailed test’s p-value by *halving* the p-value, not multiplying by 2 (as recommended above).

February 6, 2015 at 5:05 pm

Thanks, Paul. Yes. I fixed the half.

While it’s true that F-tests are one-tailed, they’re not testing directional hypotheses, the way a one-tailed t or z test does.

October 29, 2014 at 11:15 pm

I trying to complete the results of my study and need to know how to convert my 1-tailed results to 2-tailed? The company that ran my stats used a 1-tailed instead of 2-tailed, which as I understand is what I should have used to show directionality. Nicole

November 3, 2014 at 4:41 pm

Hi Nicole, just double them.

September 26, 2014 at 8:13 am

Very interesting. I am reading a much celebrated book (The weakness of Civil Society in Post-Communist Europe, by Marc Howard, 2003) in polticial science at the moment containing regression analysis with one tailed coeficients. This, togheter with that they only have a N of 23 (with 5 independent variables), raise my eyebrows. The results are also quite controversial….

What would you off-hand say about that?

September 26, 2014 at 12:27 pm

Without knowing anything else, the one tailed tests of coefficients wouldn’t worry me too much except for the fact you said it’s controversial. Which means perhaps the opposite result is reasonable.

The N of 23 is more of a concern. That’s pretty small. I find results like this are not bad per se. It’s fine to consider as one possible piece of information–they are great for spurring more research. But you can’t make any conclusions based on them.

June 14, 2013 at 9:54 pm

Jst wanna thank you for your post ; it will save me on my exam tomorrow (y)

& in my course it is not always divided by 2 (p-value). It depends on the value of your t (,= 0) & if your H1:’value” is > or 0, Hasard ?

Even if i’m wrong, I want to thank you again !

January 2, 2013 at 8:16 am

how can we calculate p-value of one-tailed from two-tailed hypothesis in spss?

January 2, 2013 at 10:50 am

All you have to do is divide it by 2.

December 1, 2012 at 11:04 am

i found this issue more important please continue in such way. kind regards, nurilign.

August 27, 2010 at 12:59 pm

Thank you so much for your reply and offer to try a find a source regarding the limited utility of one-tailed tests when doing ANOVAs and regressions, as well as the advice for converting two-tailed tests to one-tailed tests.

The only problem I have is that the Pearson Correlation Coefficient output from my stats consultant does not contain the p value. In order to calculate the p value for a two-tailed test, I thought it might be possible to take the df (n-2 for two-tailed tests) and look up the significance level in the table of critical values of the correlation coefficient. Once I get those values, I would simply divide by 2 to get the one-tailed level of significance. Do you think that is a statistically sound procedure.

Thank you again for your assistance, Sue

August 25, 2010 at 11:29 am

I am currently working on my dissertation and one of my committee members suggested that I should have used a one-tailed test as I have a directional hypothesis, but I think that a two-tailed test is just as appropriate based on several of the reasons listed on the blog.

I was particularly intrigued by the statement that “F tests, chi-square tests, etc. can’t accommodate one-tailed tests because their distributions are not symmetric.” This would make a fine argument for not re-rerunning my data and was wondering if there is a reference or citation for that point. I have not been able to find that point in any of the stats texts that I own. Any help would be greatly appreciated!

August 26, 2010 at 7:23 pm

Hi Sue, Hmmm. I would think that texts that talk about the F-test would mention that it's not symmetric. You could certainly use any text that states that t-squared=F. But I'll see if I can find something that says it directly. But in any case, you don't have to rerun anything, even if you weren't using F tests. To get a one-sided p-value, just halve the two-sided p-value you have. Karen

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• Key Differences

Know the Differences & Comparisons

Difference Between One-tailed and Two-tailed Test

To test the hypothesis, test statistics is required, which follows a known distribution. In a test, there are two divisions of probability density curve, i.e. region of acceptance and region of rejection. the region of rejection is called as a critical region .

In the field of research and experiments, it pays to know the difference between one-tailed and two-tailed test, as they are quite commonly used in the process.

Content: One-tailed Test Vs Two-tailed Test

Comparison chart, definition of one-tailed test.

One-tailed test alludes to the significance test in which the region of rejection appears on one end of the sampling distribution. It represents that the estimated test parameter is greater or less than the critical value. When the sample tested falls in the region of rejection, i.e. either left or right side, as the case may be, it leads to the acceptance of alternative hypothesis rather than the null hypothesis. It is primarily applied in chi-square distribution; that ascertains the goodness of fit.

In this statistical hypothesis test, all the critical region, related to α , is placed in any one of the two tails. One-tailed test can be:

• Left-tailed test : When the population parameter is believed to be lower than the assumed one, the hypothesis test carried out is the left-tailed test.
• Right-tailed test : When the population parameter is supposed to be greater than the assumed one, the statistical test conducted is a right-tailed test.

Definition of Two-tailed Test

The two-tailed test is described as a hypothesis test, in which the region of rejection or say the critical area is on both the ends of the normal distribution. It determines whether the sample tested falls within or outside a certain range of values. Therefore, an alternative hypothesis is accepted in place of the null hypothesis, if the calculated value falls in either of the two tails of the probability distribution.

In this test, α is bifurcated into two equal parts, placing half on each side, i.e. it considers the possibility of both positive and negative effects. It is performed to see, whether the estimated parameter is either above or below the assumed parameter, so the extreme values, work as evidence against the null hypothesis.

Key Differences Between One-tailed and Two-tailed Test

The fundamental differences between one-tailed and two-tailed test, is explained below in points:

• One-tailed test, as the name suggest is the statistical hypothesis test, in which the alternative hypothesis has a single end. On the other hand, two-tailed test implies the hypothesis test; wherein the alternative hypothesis has dual ends.
• In the one-tailed test, the alternative hypothesis is represented directionally. Conversely, the two-tailed test is a non-directional hypothesis test.
• In a one-tailed test, the region of rejection is either on the left or right of the sampling distribution. On the contrary, the region of rejection is on both the sides of the sampling distribution.
• A one-tailed test is used to ascertain if there is any relationship between variables in a single direction, i.e. left or right. As against this, the two-tailed test is used to identify whether or not there is any relationship between variables in either direction.
• In a one-tailed test, the test parameter calculated is more or less than the critical value. Unlike, two-tailed test, the result obtained is within or outside critical value.
• When an alternative hypothesis has ‘≠’ sign, then a two-tailed test is performed. In contrast, when an alternative hypothesis has ‘> or <‘ sign, then one-tailed test is carried out.

To sum up, we can say that the basic difference between one-tailed and two-tailed test lies in the direction, i.e. in case the research hypothesis entails the direction of interrelation or difference, then one-tailed test is applied, but if the research hypothesis does not signify the direction of interaction or difference, we use two-tailed test.

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Swati Aggarwal says

April 24, 2018 at 11:47 am

Very Informative and specifically summarised. thank you.

Aurobindo says

January 20, 2019 at 8:22 am

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July 23, 2020 at 11:48 pm

This website is very useful and easy to understand for Statistics methods and concepts.

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January 11, 2023 at 5:13 am

The website is very linear to a layman statistics. I really enjoyed visiting the site

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What Is a One-Tailed Test?

• Determining Significance
• One-Tailed Test FAQs
• Corporate Finance
• Financial Analysis

One-Tailed Test Explained: Definition and Example

Investopedia / Xiaojie Liu

A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis.

Financial analysts use the one-tailed test to test an investment or portfolio hypothesis.

Key Takeaways

• A one-tailed test is a statistical hypothesis test set up to show that the sample mean would be higher or lower than the population mean, but not both.
• When using a one-tailed test, the analyst is testing for the possibility of the relationship in one direction of interest and completely disregarding the possibility of a relationship in another direction.
• Before running a one-tailed test, the analyst must set up a null and alternative hypothesis and establish a probability value (p-value).

A basic concept in inferential statistics is hypothesis testing . Hypothesis testing is run to determine whether a claim is true or not, given a population parameter. A test that is conducted to show whether the mean of the sample is significantly greater than and significantly less than the mean of a population is considered a two-tailed test . When the testing is set up to show that the sample mean would be higher or lower than the population mean, it is referred to as a one-tailed test. The one-tailed test gets its name from testing the area under one of the tails (sides) of a normal distribution , although the test can be used in other non-normal distributions.

Before the one-tailed test can be performed, null and alternative hypotheses must be established. A null hypothesis is a claim that the researcher hopes to reject. An alternative hypothesis is the claim supported by rejecting the null hypothesis.

A one-tailed test is also known as a directional hypothesis or directional test.

Example of the One-Tailed Test

Let's say an analyst wants to prove that a portfolio manager outperformed the S&P 500 index in a given year by 16.91%. They may set up the null (H 0 ) and alternative (H a ) hypotheses as:

H 0 : μ ≤ 16.91

H a : μ > 16.91

The null hypothesis is the measurement that the analyst hopes to reject. The alternative hypothesis is the claim made by the analyst that the portfolio manager performed better than the S&P 500. If the outcome of the one-tailed test results in rejecting the null, the alternative hypothesis will be supported. On the other hand, if the outcome of the test fails to reject the null, the analyst may carry out further analysis and investigation into the portfolio manager’s performance.

The region of rejection is on only one side of the sampling distribution in a one-tailed test. To determine how the portfolio’s return on investment compares to the market index, the analyst must run an upper-tailed significance test in which extreme values fall in the upper tail (right side) of the normal distribution curve. The one-tailed test conducted in the upper or right tail area of the curve will show the analyst how much higher the portfolio return is than the index return and whether the difference is significant.

1%, 5% or 10%

The most common significance levels (p-values) used in a one-tailed test.

Determining Significance in a One-Tailed Test

To determine how significant the difference in returns is, a significance level must be specified. The significance level is almost always represented by the letter p, which stands for probability. The level of significance is the probability of incorrectly concluding that the null hypothesis is false. The significance value used in a one-tailed test is either 1%, 5%, or 10%, although any other probability measurement can be used at the discretion of the analyst or statistician. The probability value is calculated with the assumption that the null hypothesis is true. The lower the p-value , the stronger the evidence that the null hypothesis is false.

If the resulting p-value is less than 5%, the difference between both observations is statistically significant, and the null hypothesis is rejected. Following our example above, if the p-value = 0.03, or 3%, then the analyst can be 97% confident that the portfolio returns did not equal or fall below the return of the market for the year. They will, therefore, reject H 0  and support the claim that the portfolio manager outperformed the index. The probability calculated in only one tail of a distribution is half the probability of a two-tailed distribution if similar measurements were tested using both hypothesis testing tools.

When using a one-tailed test, the analyst is testing for the possibility of the relationship in one direction of interest and completely disregarding the possibility of a relationship in another direction. Using our example above, the analyst is interested in whether a portfolio’s return is greater than the market’s. In this case, they do not need to statistically account for a situation in which the portfolio manager underperformed the S&P 500 index. For this reason, a one-tailed test is only appropriate when it is not important to test the outcome at the other end of a distribution.

How Do You Determine If It Is a One-Tailed or Two-Tailed Test?

A one-tailed test looks for an increase or decrease in a parameter. A two-tailed test looks for change, which could be a decrease or an increase.

What Is a One-Tailed T Test Used for?

A one-tailed T-test checks for the possibility of a one-direction relationship but does not consider a directional relationship in another direction.

When Should a Two-Tailed Test Be Used?

You would use a two-tailed test when you want to test your hypothesis in both directions.

University of Southern California. " FAQ: What Are the Differences Between One-Tailed and Two-Tailed Tests? "

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Difference Between One-Tailed and Two-Tailed Tests

• Differences Between two-sample, t-test and paired t-test
• Difference Between Test E And Test C
• Difference between End-to-end Testing and Unit Testing
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One and Two-Tailed Tests are ways to identify the relationship between the statistical variables. For checking the relationship between variables in a single direction (Left or Right direction), we use a one-tailed test. A two-tailed test is used to check whether the relations between variables are in any direction or not.

One-Tailed Test

A one-tailed test is based on a uni-directional hypothesis where the area of rejection is on only one side of the sampling distribution. It determines whether a particular population parameter is larger or smaller than the predefined parameter. It uses one single critical value to test the data.

Alternative Hypothesis ( H 1​):

• For a right-tailed test: *** QuickLaTeX cannot compile formula: H_1: \theta > \theta_0 *** Error message: Cannot connect to QuickLaTeX server: cURL error 52: Empty reply from server Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37
• For a left-tailed test: *** QuickLaTeX cannot compile formula: H_1: \theta < \theta_0 *** Error message: Cannot connect to QuickLaTeX server: cURL error 52: Empty reply from server Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

Test Statistic: Depending on the type of test and the distribution, the test statistic is computed ( Z -score for normal distribution).

Decision Rule: If the test statistic falls in the critical region, reject the null hypothesis in favor of the alternative hypothesis.

Example: Effect of participants of students in coding competition on their fear level.

• H0: There is no important effect of students in coding competition on their fear level. 

The main intention is to check the decreased fear level when students participate in a coding competition.

Two-Tailed Test

A two-tailed test is also called a nondirectional hypothesis. For checking whether the sample is greater or less than a range of values, we use the two-tailed. It is used for null hypothesis testing.

Alternative Hypothesis ( H 1​): *** QuickLaTeX cannot compile formula: \text{Alternative Hypothesis } (H_1): \theta \neq \theta_0 \text{ (not equal to)} *** Error message: Cannot connect to QuickLaTeX server: cURL error 52: Empty reply from server Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

Test Statistic: Compute the test statistic as appropriate for the distribution ( Z -score for normal distribution).

Decision Rule: If the test statistic falls in either tail of the distribution’s critical region, reject the null hypothesis in favor of the alternative hypothesis.

Example: Effect of new bill pass on the loan of farmers. 

• H0: There is no significant effect of the new bill passed on loans of farmers.

New bill passes can affect in both ways either increase or decrease the loan of farmers.

Difference Between One and Two-Tailed Test:

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6.4: One- and Two-Tailed Tests

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Learning Objectives

• Define Type I and Type II errors
• Interpret significant and non-significant differences
• Explain why the null hypothesis should not be accepted when the effect is not significant

In the James Bond case study, Mr. Bond was given \(16\) trials on which he judged whether a martini had been shaken or stirred. He was correct on \(13\) of the trials. From the binomial distribution, we know that the probability of being correct \(13\) or more times out of \(16\) if one is only guessing is \(0.0106\). Figure \(\PageIndex{1}\) shows a graph of the binomial distribution. The red bars show the values greater than or equal to \(13\). As you can see in the figure, the probabilities are calculated for the upper tail of the distribution. A probability calculated in only one tail of the distribution is called a "one-tailed probability."

Binomial Calculator

A slightly different question can be asked of the data: "What is the probability of getting a result as extreme or more extreme than the one observed?" Since the chance expectation is \(8/16\), a result of \(3/16\) is equally as extreme as \(13/16\). Thus, to calculate this probability, we would consider both tails of the distribution. Since the binomial distribution is symmetric when \(\pi =0.5\), this probability is exactly double the probability of \(0.0106\) computed previously. Therefore, \(p = 0.0212\). A probability calculated in both tails of a distribution is called a "two-tailed probability" (see Figure \(\PageIndex{2}\)).

Should the one-tailed or the two-tailed probability be used to assess Mr. Bond's performance? That depends on the way the question is posed. If we are asking whether Mr. Bond can tell the difference between shaken or stirred martinis, then we would conclude he could if he performed either much better than chance or much worse than chance. If he performed much worse than chance, we would conclude that he can tell the difference, but he does not know which is which. Therefore, since we are going to reject the null hypothesis if Mr. Bond does either very well or very poorly, we will use a two-tailed probability.

On the other hand, if our question is whether Mr. Bond is better than chance at determining whether a martini is shaken or stirred, we would use a one-tailed probability. What would the one-tailed probability be if Mr. Bond were correct on only \(3\) of the \(16\) trials? Since the one-tailed probability is the probability of the right-hand tail, it would be the probability of getting \(3\) or more correct out of \(16\). This is a very high probability and the null hypothesis would not be rejected.

The null hypothesis for the two-tailed test is \(\pi =0.5\). By contrast, the null hypothesis for the one-tailed test is \(\pi \leq 0.5\). Accordingly, we reject the two-tailed hypothesis if the sample proportion deviates greatly from \(0.5\) in either direction. The one-tailed hypothesis is rejected only if the sample proportion is much greater than \(0.5\). The alternative hypothesis in the two-tailed test is \(\pi \neq 0.5\). In the one-tailed test it is \(\pi > 0.5\).

You should always decide whether you are going to use a one-tailed or a two-tailed probability before looking at the data. Statistical tests that compute one-tailed probabilities are called one-tailed tests; those that compute two-tailed probabilities are called two-tailed tests. Two-tailed tests are much more common than one-tailed tests in scientific research because an outcome signifying that something other than chance is operating is usually worth noting. One-tailed tests are appropriate when it is not important to distinguish between no effect and an effect in the unexpected direction. For example, consider an experiment designed to test the efficacy of a treatment for the common cold. The researcher would only be interested in whether the treatment was better than a placebo control. It would not be worth distinguishing between the case in which the treatment was worse than a placebo and the case in which it was the same because in both cases the drug would be worthless.

Some have argued that a one-tailed test is justified whenever the researcher predicts the direction of an effect. The problem with this argument is that if the effect comes out strongly in the non-predicted direction, the researcher is not justified in concluding that the effect is not zero. Since this is unrealistic, one-tailed tests are usually viewed skeptically if justified on this basis alone.

Statistics Made Easy

Two-Tailed Hypothesis Tests: 3 Example Problems

In statistics, we use hypothesis tests to determine whether some claim about a population parameter is true or not.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter = ≤, ≥ some value

H A (Alternative Hypothesis): Population parameter <, >, ≠ some value

There are two types of hypothesis tests:

• One-tailed test : Alternative hypothesis contains either < or > sign
• Two-tailed test : Alternative hypothesis contains the ≠ sign

In a two-tailed test , the alternative hypothesis always contains the not equal ( ≠ ) sign.

This indicates that we’re testing whether or not some effect exists, regardless of whether it’s a positive or negative effect.

Check out the following example problems to gain a better understanding of two-tailed tests.

Example 1: Factory Widgets

Suppose it’s assumed that the average weight of a certain widget produced at a factory is 20 grams. However, one engineer believes that a new method produces widgets that weigh less than 20 grams.

To test this, he can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ = 20 grams
• H A (Alternative Hypothesis): μ ≠ 20 grams

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The engineer believes that the new method will influence widget weight, but doesn’t specify whether it will cause average weight to increase or decrease.

To test this, he uses the new method to produce 20 widgets and obtains the following information:

• n = 20 widgets
• x = 19.8 grams
• s = 3.1 grams

Plugging these values into the One Sample t-test Calculator , we obtain the following results:

• t-test statistic: -0.288525
• two-tailed p-value: 0.776

Since the p-value is not less than .05, the engineer fails to reject the null hypothesis.

He does not have sufficient evidence to say that the true mean weight of widgets produced by the new method is different than 20 grams.

Example 2: Plant Growth

Suppose a standard fertilizer has been shown to cause a species of plants to grow by an average of 10 inches. However, one botanist believes a new fertilizer causes this species of plants to grow by an average amount different than 10 inches.

To test this, she can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ = 10 inches
• H A (Alternative Hypothesis): μ ≠ 10 inches

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The botanist believes that the new fertilizer will influence plant growth, but doesn’t specify whether it will cause average growth to increase or decrease.

To test this claim, she applies the new fertilizer to a simple random sample of 15 plants and obtains the following information:

• n = 15 plants
• x = 11.4 inches
• s = 2.5 inches
• t-test statistic: 2.1689
• two-tailed p-value: 0.0478

Since the p-value is less than .05, the botanist rejects the null hypothesis.

She has sufficient evidence to conclude that the new fertilizer causes an average growth that is different than 10 inches.

Example 3: Studying Method

A professor believes that a certain studying technique will influence the mean score that her students receive on a certain exam, but she’s unsure if it will increase or decrease the mean score, which is currently 82.

To test this, she lets each student use the studying technique for one month leading up to the exam and then administers the same exam to each of the students.

She then performs a hypothesis test using the following hypotheses:

• H 0 : μ = 82
• H A : μ ≠ 82

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The professor believes that the studying technique will influence the mean exam score, but doesn’t specify whether it will cause the mean score to increase or decrease.

To test this claim, the professor has 25 students use the new studying method and then take the exam. He collects the following data on the exam scores for this sample of students:

• t-test statistic: 3.6586
• two-tailed p-value: 0.0012

Since the p-value is less than .05, the professor rejects the null hypothesis.

She has sufficient evidence to conclude that the new studying method produces exam scores with an average score that is different than 82.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing What is a Directional Hypothesis? When Do You Reject the Null Hypothesis?

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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